====== PPSC Paper 2021 (Lecturer in Mathematics) ====== {{ :ppsc:ppsc-maths-paper-2021.jpg|PPSC Paper 2011 (Lecturer in Mathematics)}} On this page, we have given question from old (past) paper of Lecturer in Mathematics conducted in year 2021. This is a MCQs paper and answers are given at the end of the paper. At the end of the PDF is also given to download. This paper is provided by Ms. [[:people:iqra-liaqat]]. We are very thankful to her for providing this paper. - How many women candidates won National Assembly seats directly in General Election \(2018\)? - $4$ - \(6\) - $8$ - $10$ - Let \(X\) and \(Y\) be Banach spaces. Then the product space \(X\times Y\), with the norm defined by: \(\parallel (x,y) \parallel=\parallel x\parallel+\parallel y\parallel, \,\forall \, (x,y)\in X \times Y\) \\ - Banach space - Normed space - Linear space - All of these - Let \(f(x)=[x],\) greatest integer \(\leq x;\) be integrable function on \([0,4]\), then \(\displaystyle \int_{0}^{4}[x]dx \) is equal to: \\ - \(3\) - \(7\) - \(6\) - \(4\) - Solution of \((\bigtriangleup ^2=2\bigtriangleup +1) u_x=3x+2\) is: \\ - \(u_x=3x+4\) - \(u_x=4x+3\) - \(u_x=3x-4\) - \(u_x=4x-3\) - The sequence \(\left\{\dfrac{2ni}{n+i}-\dfrac{(9-12i)n+2}{3n+1+7i}\right\}\) converges to : - \(3+6i\) - \(-3-6i\) - \(-3+6i\) - \(3-6i\) - Which of those is not an analytical method to solve partial differential equation? \\ - Change of variable - Superposition principle - Finite element method - Integral transform - Order of convergence of Newton's method is : \\ - Quadratic - Cubic - $4^{th}$ - Undefined - After discretizing the partial differential equations take which if these forms? \\ - Exponential equations - Trigonometric equations - Logarithmic equations - Algebraic equations - For a function \(f\), if \(f_{xx}=f_{xy}=f_{yy}=0\), the point \((x,y)\) will be multiple point of order: \\ - Lower than two - Two - Higher - Higher than the two - Suppose that \(X\) and \(Y\) are closed subspaces of a Hilbert space \(H\) such that \(X \perp Y\), then \(X+Y\) is : \\ - A closed subspace - \(X^\perp +Y^\perp\) - \(X^\perp + Y\) - Normed subspace - The function \(f(x)=x^{(-1)}\) is not: \\ - Uniform continuous on \((0,1)\) - Continuous on \((0,1)\) - Differentiable on \([0,1)\) - Both \(A\) and \(B\) - ${\rm Ln} z$ is discontinuous on axis: \\ - Positive real - Nonpositive real - Negative - Nonnegative - Normal component of an acceleration is : \\ - $v/p$ - $v^2/p$ - $p^2/v$ - $p/v$ - Every metric space is a : \\ - Hausdroff space - $T_2$ space - Both \(A\) and \(B\) - $T_3$ - \(\mathbb{N}'\), derived set of \(\mathbb{N}\), is: \\ - \(\phi \) - \(\mathbb{R}\) - \(\mathbb{Q}\) - Not exist - Given \( \left(a_n\right)_{n\in \mathbb{N}}\), where \(a_n>0 \,\forall \, n\). If \(\lim\limits_{n \to \infty} a_n=l>0\), then \(\lim\limits_{n \to \infty}(a_n\ldots a_1)^{\frac{1}{n}}\) is - \(l^n\) - \(1/l\) - \(l\) - \(1/l^n\) - If \(\vec{F}\) is a continuously differentiable vector point function and \(V\) is the volume bounded by a closed surface \(S\), then \(\displaystyle \iint_{c}\vec{F}\times \vec{n}dS=\iiint_{c}{\rm div} \vec{F} dV\) is called - Gauss' divergence theorem - Surface integral - Volume integral - None of these - \(c_{ijk}c_{ijk}\) is equal : \\ - $4$ - $3$ - $6$ - \(2\) - \(\displaystyle \sum_{d|n} 1/d=\sigma(n)/n\) for each integer: \\ - \(n\geq 1\) - \(n\geq 3\) - \(n\geq 7\) - None of these - Let \(A\) and \(B\) be two non-square matrices such that \(AB=A\) and \(BA=B\), then \(A\) and \(B\) are matrices:\\ - Idempotent - Involuntary - Nilpotent - No conclusion - The Cauchy-Riemann equations can be satisfied at a point $z$, but the function $f(z)=u(x,y)+iv(x,y)$ can be at $z$: \\ - Differentiable - Non-differentiable - Continuous - None of these - Let $G$ and $G'$ be two groups. Then a homomorphism $f:G\to G'$ is one-one iff: \\ - \(\text{Ker } f=\{e\} \) - \(\text{Im } f=\{e\}\) - $\text{Ker } f \neq \{e\}$ - \(\text{Im } f=\{e\} \) - The sequence $\left(1 / n^2 \right)_{n\in \mathbb{N}}$ is: \\ - convergent - Cauchy - Bounded - Both A \& B - Kronecker delta is a tensor of rank : \\ - \(3\) - \(2\) - \(1\) - any - Let \(G\) be a group and \(a\in G\) is of finite order \(n\) such that \(a^m=0\), then \\ - \(m |n\) - \(n|m\) - \((m,n)=1\) - \(n\) - \(4\times \) volume of a tetrahedron is equal to the volume of: \\ - Parallelepiped - Cube - Cuboid - None - Let \(A\) be the real matrix with the rows from an orthonormal set, then \(A\) is : \\ - Normal - Orthogonal - Column of \(A\) from an orthonormal set - Both \(A\) and \(C\) - The solution \(\sin2x\) and \(\cos2x\) of the differential equation \(y''+4y=0\) are \\ - Independent - Dependent - Wronskian of both is zero - Both \(A\) and \(C\) - Let \(V\) be an inner product space and \(u,v\in V\), Then \(|\textless u,v\textgreater|=||u||\,||v||\) iff: \\ - \(u\) nd \( v\) are linearly independent - \(u\) and \( v\) are linearly dependent - \(u \) and \( v\) are scalar multiple of each other - Both \(B\) and \(C\) - Let \(U \) \& \(V\) be two vectors spaces such that \(T:V\to U\), a linear transformation. Then : \\ - \(\dfrac{V}{{\rm Ker} T}\cong U\) - \( \dfrac{V}{{\rm Ker} T}\cong V\) - \(\dfrac{U}{{\rm Ker} T}\cong V\) - \(\dfrac{V}{{\rm Ker} T}\cong T(V)\) - Let $x,p\in \mathbb{R}$, $x+1>0$, $p\neq 0,1$ be such that \((1+x)^p<1+px\), then: \\ - $0\leq p\leq 1$ - $0\leq p< 1$ - $0< p\leq 1$ - $0< p< 1$ - The Diophantine equation \(15x+51y=14\) has solution: \\ - Unique - More than one - No - Arbitrary - Kepler stated the first law of planetary motion in: \\ - \(1709 \) - \(1609 \) - \(1507\) - \(1607 \) - Rings of integers has characteristic: \\ - \(1\) - \(0\) - \(\infty\) - \(-1\) - Every invertible diagonal matrix is a matrix: \\ - Scalar - Lower triangular - Upper triangular - Both \(B\) \& \(C\) - \(\tanh^{-1}z\) is not defined for \(z\) equal to: \\ - $1$ - \(-1\) - \(\pm 1\) - Complex plane - Every homogenous system of linear equations has solution: \\ - Trivial - Non-trivial - Parametric - None of these - Let \(H\) be a subgroup of a group \(G\) such that \(Ha\neq Hb\), then: \\ - \(aH=bH\) - \(aH \subseteq bH\) - \(bH \subseteq aH\) - \(aH\neq bH\) - If \(S_1,S_2\) are subsets of \(V(F)\) and \(L(S_1)\) is the linear space of \(S_1\), then: \\ - \( L(L(S_1))=L(S_1)\) - \( L(S_1\cup S_2)=L(S_1)+L(S_2)\) - Both \(A\) and \(B\) - \( L(S_1)\subseteq L(S_2)\) - The summation index is also called: \\ - Dummy index - Free index - Convention - Both \(A\) and \(B\) - Reduced echelon form of a matrix is: \\ - Unique - Not unique - Pivot element are 1 - Both A and C - The number of asymptotes of an algebraic curve of the $n$th degree: \\ - Exceed \(n\) - Cannot exceed \(n\) - Exactly \(n\) - Both $A$ and \(C\) - Let $H$, $K$ be subgroups of a group \(G\), Then \(HK\) is a subgroup of \(G\) iff: \\ - \(HK = KH\) - \(HK \neq KH\) - \(H^{-1}=K^{-1}\) - \((HK)^{-1}=K^{-1}H^{-1}\) - If \(A\) and \(B\) are two ideals of a ring \(R\), Then \(A+B\) is an ideal of \(R\) containing: \\ - \(A\) - \(B\) - Both $A$ \& \(C\) - None of these - A normed space \(X\) is finite dimension iff \(X\) is: \\ - Compact - Connected - Locally compact - Homeomorphic - The symbol $A_{ijk}$, $\{i,j,k=1,2,3\}$ denotes numbers: \\ - \(27\) - \(9\) - \(8\) - \(4\) - The function \(f(z)\) is analytical in a domain \(D\) and \(f(z)=c+iv(x,y)\), where \(c\) is a real constant. Then \(f\) in \(D\) is a: \\ - Constant - Nonconstant - Continuous - None of these - The centre of curvature at any point \(P\) of a curve is the point which on the positive direction of the normal at \(P\) and is at a distance \\ - \(x\,(keps)\) - \(\ell(rho)\) - \(\dfrac{1}{x}\) - \(\dfrac{1}{\ell}\) - Time of flight of the projectile is: \\ - \(\dfrac{2v_0 \sin\alpha}{-g}\) - \(\dfrac{2v_0 \cos\alpha}{g}\) - \(\dfrac{v_0 \sin2\alpha \sec \alpha}{g}\) - \(\dfrac{v_0 \sin2\alpha}{-g\cos\alpha}\) - For a function \(f(x,y)\) in a region \(D\) in \(xy\) plane, the condition \(|f(x,y_2)-f(x,y_1)|\leq K|y_2-y_1|\) is called Lipschitz, provided that: \\ - \(K=0\) - \(K>0\) - \(K<0\) - \(K\in \mathbb{R}\) - If \(k\) integers \(a_1,a_2,...,a_k\) form a complete residue system modulo \(m\), then: \\ - \(m0\) - \(f'(x)<0\) - Both A and B - \(f'(x=0)\) - The mapping \(w=z^2+1\) is conformable at: \\ - $z=-1$ - $z=1$ - $z=\pm1$ - None of these - The order of the continuity equation of steady two-dimensional flow is: \\ - \(1\) - \(0\) - \(2\) - \(3\) - Let \(a\) and \(m>0\) be integers with \(a^{\phi(m)}\equiv1 \pmod m\) provided that: \\ - $a>m$ - $m\dfrac{x}{\sin x}\) is true for: \\ - $0\leq x \leq \dfrac{\pi}{2}$ - $0< x< \dfrac{\pi}{2}$ - $0\leq x< \dfrac{\pi}{2}$ - $0< x\leq \dfrac{\pi}{2}$ - While solving a partial differential equation using a variable separable method, we equate the ratio to a constant? \\ - Can be positive or negative integer or zero - Can be positive or negative number or zero - Must be a positive integer - Must be negative integer - A particle of mass \(m\) moves in a circle of radius \(r\) with constant speed \(v\) and \(F\) an acting force, then: \\ - \(F \propto \frac{mv^2}{r}\) - \(F \propto \frac{mv}{r}\) - \(F \propto\frac{(mv^2)}{\sqrt{r}}\) - None of these - Let \(W(F)\) be a subspace of a finite dimensional vector space \(V(F)\), then \(\dim(V/W)\) is : \\ - $\dim V-\dim W$ - $\dim V+\dim W$ - $\dim V+\dim W-\dim (V \cap W)$ - $\dim V-\dim W+\dim (V \cap W)$ - If $L$,$M$ and $N$ are three subspaces of a vector space $V$ such that \(M\subseteq L\), then: \\ - $L \cap (M+N)=(L\cap M)+(L\cap N)$ - $L \cap (M+N)=M+(L\cap N)$ - $L \cup (M+N)=(L\cup M)+(L\cup N)$ - Both A and B - Changes in sign but not in magnitude when the cyclic order is changed is possible in: \\ - Vector triple product - Scalar triple product - Mixed product - Both B and C - Let \(A\) be a subspace of a topological space \(X\); let \(\bar{A}\) be its closure, then \(\bar{A}\) is equal: (provided that $A^\circ$, $b(A)$ and $A'$ are respectively interior, boundary and set of accumulation points of \(A\) respectively) \\ - $A^\circ \cup b(A)$ - $A \cup A'$ - Both A and B - $A\cap A'$ - Let \(\vec{f}(x,y,z)\) be continuously differentiable vector point function then CurlCurl\(\vec{f}+\nabla^2 f\) is: \\ - grad div\(\vec{f}\) - div grad\(\vec{f}\) - div Curl\(\vec{f}\) - Curl div\(\vec{f}\) - A square unitary matrix with real entries is: \\ - Orthogonal - Normal - None - Leslie - Every triangular matrix is: \\ - Diagonal - Lower triangular - Invertible diagonal - Both A and B - Printing press was invented by \\ - Mary Anderson - Johannes Gutenbery - George Antheil - Victor Adler - Which of the following is used for the purification of water? \\ - Oxygen - Ammonia - Chlorine - Carbon Dioxide - Faiz Ahmed Faiz was imprisoned for his alleged involvement in ----------- conspiracy. \\ - Agartala - Lahore - Attock - Rawalpindi - Turkey connects which two continents? \\ - Asia and Europe - Asia and Africa - South America and North America - Asia and Australia - Aljazeera TV channel belong to: \\ - Qatar - Kuwait - Egypt - Bahrain - In the period of which pious caliph Quranic verses were collected in one place? \\ - Hazrat Umar (RA) - Hazrat Abu Bakar (RA) - Hazrat Ali (RA) - Hazrat Usman (RA) - Given the name of the Sahabi who was given the title of Ateeq \\ - Hazrat Abu Bakar (RA) - Hazrat Umar (RA) - Hazrat Ali (RA) - Hazrat Zaid Bin Sabit (RA) - Choose the correct pronoun;- Can you please return the calculator ----- you borrow yesterday? \\ - Who - Whom - That - Whose - Fill in the suitable preposition;- "The shop is open from \(7 \)am ----- \(5\)pm. \\ - At - Until - Above - On - In computing, WAN stands for: \\ - World Area Network - Wide Area Network - World Access Network - Wireless Access Network - What is the shortcut key to hide entire column in MS Excel sheet? \\ - CTRL+O - CTRL+A - CTRL+H - CTRL+I - Which one is the capital city of Oman? \\ - Adam - Muscat - As Sib - Bahia - Tarbela is ----------- dam. \\ - Rock fill - Earth fill - Concrete - None of these - The famous pre-historic monument Stonehenge is found in ----- \\ - Greece - China - England - None of these - When India stopped supply of water to Pakistan from every canal flowing from India to Pakistan for first time after creation \\ - April 1st, 1947 - April 1st, 1948 - April 1st, 1949 - April 26, 1948 - The Pakistan International Airlines came into being in the year \\ - 1952 - 1953 - 1954 - 1955 - Sepoys in the British army raised in revolt from the city of ----- \\ - Meerut - Delhi - Lucknow - Calcutta - کے درست مطلب کا انتخاب کریں ``By Leaps and Bounds" \\ - کتے کی طرح لپکنا - جنگجو ہونا - رفتہ رفتہ آگے بڑھنا - تیزی سے ترقی کرنا - اردو میں منقوط حروف کی تعدادہے - پندرہ - سترہ - انیس - اکیس ====Answers==== 1-a, 2-b, 3-c, 4-b, 5-d, 6-b, 7-a, 8-a, 9-d, 10-a, 11-a, 12-c, 13-b, 14-c, 15-a, 16-c, 17-a, 18-b, 19-a, 20-a, 21-b, 22-a, 23-d, 24-b, 25-b, 26-a, 27-b, 28-a, 29-c, 30-d, 31-a, 32-d, 33-a, 34-b, 35-d, 36-d, 37-a, 38-b, 39-c, 40-d, 41-d, 42-b, 43-a, 44-c, 45-a, 46-a, 47-a, 48-b, 49-c, 50-b, 51-b, 52-c, 53-a, 54-c, 55-c, 56-c, 57-b, 58-c, 59-b, 60-a, 61-a, 62-c, 63-b, 64-a, 65-d, 66-c, 67-a, 68-c, 69-b, 70-c, 71-a, 72-b, 73-a, 74-a, 75-a, 76-a, 77-d, 78-c, 79-b, 80-a, 81-a, 82-b, 83-c, 84-d, 85-a, 86-a, 87-b, 88-a, 89-b, 90-b, 91-b, 92-a, 93-b, 94-b, 95-c, 96-b, 97-d, 98-a, 99-d, 100-b ==== Download ==== * **{{ :ppsc:ppsc-paper-2021-mathcity.org.pdf |Download PDF}}** * View Online {{gview noreference>:ppsc:ppsc-paper-2021-mathcity.org.pdf}} {{tag>PPSC Iqra_Liaqat}}